In NMR spectroscopy, the collective measurement is weakly invasive and its back-action is called radiation damping. The aim of this talk is to provide a control-theoretical analysis of the problem of suppressing this radiation damping. It is shown that the two feedback schemes commonly used in the NMR practice correspond one to a high gain oputput feedback for the simple case of maintaining the spin 1/2 in its inverted state, and the second to a 2-degree of freedom control design with a prefeedback that exactly cancels the radiation damping field. A general high gain feedback stabilization design not requiring the knowledge of the radiation damping time constant is also investigated.

Interaction of ultrashort intense laser pulses with molecular media leads to highly nonlinear nonperturbative effects which can only be treated by large scale computation on massively parallel computers. Single molecule response to such pulses leads to Molecular High Order Harmonic Generation, MHOHG, (1), from which one can synthesize new "attosecond" pulses necessary to control electron dynamics at the natural time scale of the electron, the attoseocond (10**-18 s), (2).The single molecular response can be obtained from high level quantum Time-Dependent Schrödinger,TDSE, simulations. The collective macroscopic response of a molecular medium requires solving many TDSE,s (>10**5)coupled to the classical laser (photon) Maxwell equations (3). We will present the numerical methods necessary to achieve this goal, especially the problem of transparent and artificial boundary condition techniques in view of the different time scales, photon vs electron. Results will be shown for attosecond pulse generation and pulse filamentation in an aligned molecular medium, the one electron H2+ system(4).

(1).A D Bandrauk et al,"Molecular Harmonic Generation," in Progress in Ultrafast Intense Laser Science, vol III, edit K. Yamanouchi (Springer V, NY,2008), chapt 9. (2).A D Bandrauk,F Krausz, A Starace, "Focus on Attosecond Physics," New J Phys, 10, 025004(2008). (3).E Lorin,S Chelkowski, A D Bandrauk,"Maxwell-Schrödinger Equations for Nonlinear Laser Propagation in Molecular Media," Comput. Phys. Commun. 177, 908 (2007). (4).E Lorin,S Chelkowski,A D bandrauk,"Attosecond Pulse Generation for Aligned Molecules," New J Phys, 10, 025033(2008).

In this talk we study the dissipative dynamics arising from coupling to an infinite field in both the classical and quantum context. In particular we study dissipative dynamics generalizing the classical Lamb model. We apply this to the study of dissipation arising in certain controlled quantum systems and also study a model which allows us to quantize certain nonholonomic systems. In the latter case we consider nonholonomic constraints as arising from the limit of a frictional force and then implement the force by an external field which we then quantize. Other methods of quantizing nonholonomic systems will also be discussed.

Realistic quantum mechanical systems are influenced through the coupling to an environment containing a large number of mostly uncontrollable degrees of freedom. This unavoidable interaction of an open quantum systems with its environment leads to the mechanisms of dissipation and damping, and to a strong and often rapid loss of quantum coherence. The talk begins with a brief introduction into the standard theory of quantum mechanical relaxation which is based on the Markov approximation and on the concepts of completely positive dynamical semigroups and of quantum master equations in Lindblad form. Many examples for this approach are known from quantum optics, decoherence theory, quantum Brownian motion and quantum measurement and control theory. However, strong couplings or interactions with low-temperature reservoirs generally lead to large system-environment correlations which result in long memory times and in a failure of the Markov approximation. To describe the basic features of the non-Markovian quantum dynamics of open systems we develop several new methods as, for example, the technique of correlated projection superoperators [1] and the concept of quantum semi-Markov processes [2]. A number of examples and applications to structured and finite reservoirs [3], to electron spin dynamics in quantum dots [4], and to the problem of quantum transport in nano-structures [5] will be discussed.

[1] H. P. Breuer, Phys. Rev. A 75, 022103 (2007).

[2] H. P. Breuer and B. Vacchini, Phys. Rev. Lett. 101, 140402 (2008).

[3] H. P. Breuer, J. Gemmer and M. Michel, Phys. Rev. E73, 016139 (2006).

[4] E. Ferraro, H. P. Breuer, A. Napoli, M. A. Jivulescu, and A. Messina, Phys. Rev. B78, 064309 (2008).

[5] R. Steinigeweg, H. P. Breuer and J. Gemmer, Phys. Rev. Lett. 99, 150601 (2007).

We present a recently developed mixed quantum-classical method which accounts for the evolution of a quantum subsystem coupled to a non-equilibrium environment (solvent) described in an extended hydrodynamic setting [1]. Starting from a hybrid quantum-classical phase-space distribution, coupled equations for the quantum-classical local density and momentum density are derived which feature the characteristic population-coherence coupling of the nonadiabatic quantum evolution. A generalized free energy functional is introduced, which is similar to the functionals used in dynamical density functional theory (DDFT) methods [2] but is adapted to the quantum-classical setting. The relevant functionals involve two-particle (or, more generally, n-particle) correlation functions that are constructed from state-specific microscopic solute-solvent interactions. A microscopic Marcus-type functional for polar solvation is considered as a special case. The present formulation is particularly appropriate to describe ultrafast solvation dynamics coupled with charge transfer, for example in photochemical charge transfer processes. By the explicit consideration of quantum coherence, the details of population transfer and its susceptibility to decoherence effects, become amenable to direct investigation. First numerical examples are presented [3] and the extension of the formalism beyond the free energy functional formulation are addressed, in particular in view of including non-equilibrium solvent correlations.

[1] I. Burghardt and B. Bagchi, Chem. Phys. 329, 343 (2006).

[2] B. Bagchi and A. Chandra, Adv. Chem. Phys. LXXX, 1 (1991); U. Marini Bettolo Marconi and P. Tarazona, J. Chem. Phys. 110, 8032 (1999); A. J. Archer and R. Evans, J. Chem. Phys. 121, 4246 (2004).

[3] P. Ramanathan, S. Parry, S.-L. Zhao, K. H. Hughes, and I. Burghardt, to be submitted.

In this talk we survey some methods to study the controllability of nonlinear control systems modeled by partial differential equations, namely: 1. The return method, 2. Power series expansions, 3. Quasi-static deformations. These methods will be illustrated on quantum systems.

The effect of electron-vibrational interactions on the electronic transport induced by femtosecond omega + 2omega laser fields along unbiased molecular nanojunctions is investigated. For this, the photoinduced vibronic dynamics of trans-polyacetylene oligomers coupled to macroscopic metallic leads is followed in a mean-field mixed quantum- classical approximation. A reduced description of the dynamics is obtained by introducing projective lead-molecule couplings and deriving an effective Schrödinger equation satisfied by the orbitals in the molecular region. Two possible rectification mechanisms are identified and investigated. The first one relies on near-resonance photon absorption and is shown to be fragile to the ultrafast electronic decoherence processes introduced by the wire's vibrations. The second one employs the dynamic Stark effect and is demonstrated to be highly efficient and robust to electron-vibrational interactions.

In this joint work with Gabriel Turinici, we analyse the Lyapunov trajectory tracking of the Schrödinger equation for a second order coupling operator. We present a theoretical convergence result; for situations not covered by the first theorem we propose a numerical approach and complement it with a second theoretical result.

The control problem of generating unitary transformations is especially relevant to current research in quantum information processing and computing. Control dynamical landscapes for unitary transformations is analyzed in the infinite dimensional function space of the time-dependent external field. The dynamical analysis reveals many essential geometric features of optimal control landscapes for unitary transformations, including bounds on the local landscape slope and curvature. Close examination of the curvatures at the critical points shows that the unitary transformation control landscapes are free of local traps and proper choices of the adaptation matrix may facilitate the search for optimal control fields producing desired unitary transformations, in particular, in the neighborhood of the global extrema.

A feedback control law is developed for the stochastic control problems for quantum spin systems. It is similar to the one we analyzed for the Schroedinger control system. Also, the time optimal control problem is discussed for the deterministic quantum spin system. An algorithm based on the semismooth Newton method is developed and analyzed. Numerical findings are reported for the spin half system.

Quantum systems that can be usefully partitioned into a subsystem interacting with a bath will be considered. For such systems, a quantum-classical Liouville description of the dynamics is assumed, while retaining the full quantum equilibrium structure of the system. The equations of motion may be cast in the form of a non-Markovian equation for the diagonal elements of the subsystem density matrix. The memory kernel in this equation accounts for all coherences in the system. The conditions under which the memory kernel decays rapidly as a result of averages over quantum or classical bath equilibrium structure will discussed. When such decay is rapid, it will be shown how a lift back to the full phase space results in a Markovian master equation of motion. This equation leads to a surface-hopping trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results will be illustrated by simulations of nonadiabatic chemical dynamics.

R. Grunwald and R. Kapral, J. Chem. Phys., 126, 114109 (2007). R. Grunwald, H. Kim and R. Kapral, J. Chem. Phys., 128, 164110 (2008).

An important problem in coherent spectroscopy and quantum information science is to find limits on how close an open quantum dynamical system can be driven to a target state in the presence of dissipation and decoherence. What is the optimal excitation that achieves this objective? We describe these problems in the context of the design of multidimensional NMR experiments that maximize the efficiency of transfer of coherence between coupled spins in the presence of decoherence with the goal of optimizing the sensitivity of these experiments. We present some new mathematical techniques for computing limits on how much coherence or polarization can be transferred between coupled spins in multiple spin topologies.

Joint work with Gil Katz, David Gelman, Mark Ratner and Ronnie Kosloff.

Simulation of many body quantum dynamics scales exponentially bad with the number of degrees of freedom. Many methods are devoted to obtain a restricted many body wavefunction which still are able to approximate the quantum dynamics. In the context of system bath dynamics the surrogate Hamiltonian method the dynamics is simplified by replacing the bath Hamiltonian by a simpler version which describes the bath faithfully up to a specified time. The computation task becomes even more formidable when the dynamics takes place at a finite temperature, then formally the wavefunction has to be replaced with a density operator. We present a stochastic methods which allows to describe finite temperature dynamics within a wavefunction description. The stochastic methods are applied for the initial thermal sampling. In addition the dynamical description of the bath is extended stochasticly to take care of dephasing and energy relaxation at long times. We use this method to simulate an outstanding problem in coherent control: can we obtain weak field control of a branching ratio? The model consists of a ground state and two excited state potentials. The target is to control the population in these states using phase modulation only.

Dynamical decoupling pulse sequences have been used to extend coherence times in quantum systems ever since the discovery of the spin-echo effect. But while for good reasons the nuclear magnetic resonance (NMR) community has typically been content with moderate line narrowing, in quantum computing extremely high levels of coherence are required in order to perform meaningful computational tasks. In this talk I will describe a method of recursively concatenated dynamical decoupling pulses, designed to overcome both decoherence and operational errors [1]. For bounded-strength, non-Markovian environments, such as for the spin-bath that arises in electron- and nuclear-spin based solid-state quantum computer proposals, it is strictly advantageous to use concatenated, as opposed to standard periodic dynamical decoupling pulse sequences. Namely, the concatenated scheme is both fault-tolerant and super-polynomially more efficient, at equal cost [2,3]. Preliminary experimental results on NMR of 13C in adamantene (due to Dieter Suter, Dortmund), and NMR of the 31P donor in Si (due to Steve Lyon, Princeton), demonstrating the advantages of concatenated decoupling, will also be presented. Time permitting, I will describe our recent results on the construction of a universal set of quantum logic gates whose fidelity can be kept arbitrarily high for essentially arbitrarily long times in the presence of coupling to a spin bath, by use of concatenated decoupling.

References:

[1] K. Khodjasteh and D.A. Lidar, "Fault-Tolerant Quantum Dynamical Decoupling," Phys. Rev. Lett. 95, 180501 (2005).

[2] K. Khodjasteh and D.A. Lidar, "Performance of Deterministic Dynamical Decoupling Schemes: Concatenated and Periodic Pulse Sequences," Phys. Rev. A 75, 062310 (2007).

[3] K. Khodjasteh and D.A. Lidar, "Rigorous Bounds on the Performance of a Hybrid Dynamical Decoupling-Quantum Computing Scheme," Phys. Rev. A 78, 012355 (2008).

Joint work with Julien Salomon.

In this presentation we present and illustrate a greedy algorithm that enables in a first stage to design a set of selective laser fields that can in a second stage be used to identify some unknown parameters of quantum systems for a problem of Hamiltonian Identification.

Joint work with Andrew S. Leathers (Quantum Theory Project, Departments of Chemistry and of Physics, University of Florida, Gainesville, Florida 32611, U.S.A.).

The interaction of light with a localized (primary) region in a many atom system undergoing electronic and vibrational transitions leads to energy dissipation and uctuations through both nearly instantaneous and delayed processes. A fast dissipation typically occurs due to electronic energy relaxation in the medium, while a delayed dissipation arises from vibrational energy relaxation. A theoretical and computational treatment of these phenomena has been done in terms of a reduced density matrix (RDM) satisfying a generalized Liouville-von Neumann equation.[1] Instantaneous dissipation is described by a Lindblad term containing electronic transition rates,[2] while the delayed dissipation is given by a time integral derived from the time-correlation function (TCF) of atomic displacements in the medium.[3] We consider cases where the TCF decays exponentially (fast) or as an inverse power (slowly). An initial thermal equilibrium can not be assumed when there are long lasting interactions between the primary region and the medium. We describe a general procedure that provides the optical response in this case by calculating the difference between solutions for the RDM with and without excitation by a light pulse. We present examples for slow relaxation of optical excitation in CO/Cu(001) and Ag3/Si(111).[4]

1. D. A. Micha, A. Leathers, and B. Thorndyke in "Quantum Dynamics of Complex Molecular Systems" (Springer-Verlag, 2006) D. A. Micha and I. Burghardt, eds., pp. 165-194.

2. D. A. Micha and A. Santana, J. Phys. Chem. A 2003, 107, 7311.

3. A. S. Leathers and D. A. Micha, J. Phys. Chem. A 2006, 110, 749.

4. A. S. Leather, D. A. Micha, and D. S. Kilin, "Density matrix treatment for an electronically excited adsorbate on a solid surface", to be published.

Work partly supported by the NSF of the USA, and by the Dreyfus Foundation.

In the late 1970’s Meyer and Miller (MM) [J. Chem. Phys. 70, 3214 (1979)] presented a classical Hamiltonian corresponding to a finite set of electronic states of a molecular system (i.e., the various potential energy surfaces and their couplings), so that classical trajectory simulations could be carried out treating the nuclear and electronic degrees of freedom (DOF) in an equivalent dynamical framework (i.e., by classical mechanics), thereby describing non-adiabatic dynamics in a more unified manner. Much later Stock and Thoss (ST) [Phys. Rev. Lett. 78, 578 (1997)] showed that the MM model is actually not a ‘model’, but rather a ‘representation’ of the nuclear-electronic system; i.e., were the MMST nuclear-electronic Hamiltonian taken as a Hamiltonian operator and used in the Schrödinger equation, the exact (quantum) nuclear-electronic dynamics would be obtained. In recent years various initial value representations (IVRs) of semiclassical (SC) theory have been used with the MMST Hamiltonian to describe electronically non-adiabatic processes. Of special interest is the fact that though the classical trajectories generated by the MMST Hamiltonian (and which are the ‘input’ for an SC-IVR treatment) are 'Ehrenfest trajectories', when they are used within the SC-IVR framework the nuclear motion emerges from regions of non-adiabaticity on one potential energy surface (PES) or another, and not on an average PES as in the traditional Ehrenfest model. Examples are presented to illustrate and (hopefully) illuminate this behavior.

Joint work with Oleksiy Roslyakk (Chemistry department, University of California Irvine, USA).

Nonlinear optical spectroscopy is commonly formulated semi-classically, i.e. letting a quantum material interact with classical fields. The key quantity in this approach is the nonlinear polarization, characterizing the microscopic response of the material to the incoming fields. Its calculation can be based on either the density matrix or the wave function. The former involves forward propagation in real time and is represented by double sided Feynman diagrams in Liouville space, whereas the latter requires forward and backward propagation in Hilbert space which is carried out on the Schwinger-Keldysh closed time path loop (CTPL). Such loops are extensively used in quantum field theory of non-equilibrium states, but double-sided Feynman diagrams have become a practical tool for the design and analysis of time-domain nonlinear optical experiments.

Several fundamental ambiguities which arise in the semi-classical formulation regarding the intuitive interpretation of optical signals are resolved by combining the CTPL with a quantum description of the laser fields. In nonlinear spectroscopy of single molecules, for example, the signal cannot be given in terms of a classical response functions as predicted by the semi-classical theory. Heterodyne detection can be viewed as a stimulated process and does not require a classical local oscillator. The quantum nature of the field requires the introduction of superoperator nonequilibrium Green’s functions (SNGF), which represent both response and spontaneous fluctuations of the material. This formalism allows the computation of nonlinear optical processes involving any combination of classical and quantum optical modes. Closed correlation-function expressions are derived for the combined effects of causal response and non-causal spontaneous fluctuations. Coherent three wave mixing (sum frequency generation (SFG) and parametric down conversion (PDC)) involving one and two quantum optical modes respectively, are connected to their incoherent counterparts: two-photon-induced fluorescence (TPIF) and two-photon-emitted fluorescence (TPEF).

We show how two-photon absorption and homodyne detected difference frequency generation conducted with entangled photons can be used to manipulate interference effects and select desired Liouville space pathways of matter. Recently several groups have applied entangled photon pairs in nonlinear spectroscopy (near resonance homodyne detected sum-frequency generation (SFG), two photon induced fluorescence (TPIF) and two-photon absorption (TPA). It was demonstrated that the normally quadratic scaling of the signal with the intensity of the incoming field becomes linear when using entangled photons. This indicates that the two photons effectively act as a single particle, interacting with matter within a narrow time window. This opens new ways for manipulating nonlinear optical signals and revealing new matter information otherwise erased by interference.

• Processes involving an arbitrary number of classical and quantum modes of the radiation field are treated within the same framework.
• Loop diagrams can be used to describe all incoherent and coherent (cooperative) signals.
• A unified approach is provided for both resonant and off-resonant measurements. In the latter the material enters as a parameter in an effective Hamiltonian for the field.
• Nonlinear spectroscopy conducted with resonant classical fields only accesses the causal response function. Quantum fields reveal the broader SNGF's family which carry additional information about fluctuations.
• Spectroscopy with quantum entangled fields may be described.

1. "Nonlinear Spectroscopy with Entangled Photons Manipulating Quantum Pathways of Matter," O. Rosyak, C. Marx and S. Mukamel, Phys. Rev. A. (In press, 2009).
2. "Photon Entanglement Signatures in Homodyne Detected Difference Frequency Gene," O. Roslyak and S. Mukamel, Optics Express 17, 1093 (2009).
3. "Nonlinear Optical Spectroscopy of Single, Few and Many Molecules; Nonequilibrium Green’s Function QED Approach," C.A. Marx, U. Harbola and S. Mukamel, Phys. Rev. A. 77, 022110, 2008.



4. "A Unified Description of Sum Frequency Generation, Parametric Down Conversion and Two Photon Fluoresence," O. Roslyak, C. Marx and S. Mukamel, Molecular Physics. (In press, 2009).

We develop monotonically convergent algorithms for solving typical quantum optimal control problems in chemistry and physics. They include (1) state-to-state control for a system nonlinearly interacting with a control and (2) operator pulse design under the influence of dissipation. We discuss the solution algorithms in a unified manner. As an application of the first algorithm, we consider the alignment/orientation control of diatomic molecules. The alignment is achieved through the polarizability coupling between shaped laser pulses and molecules. When the retaining of the aligned state is chosen as a physical objective, the control pulse is shown to utilize the so-called "coherent destruction of tunneling" mechanisms. This numerical observation is confirmed by using a simple analytical model. Second application is associated with (2). In quantum information processing and quantum computer, the realization of gate operations in physical systems is essential. As the operations should be done with quite high precision, optimal control approaches could be suitable tools for this purpose. We discuss the possibility through case studies such as quantum algorithm simulations and suppression of decoherence.

We analyse the possibility of control for a coupled system of Schrödinger equations on the whole real line for the harmonic oscillator modeling a single trapped ion. In fact the coupling is due to the control which acts as a potential and which is performed by three monochromatic waves which can be switched on and off, only one of them being active at each time. By taking the frequency of these waves large enough, we show that this sytem can be approximated by a much simpler one, the so-called Law-Eberly system, for which we can give an explicit control satisfying all requirements. This enables us to prove approximate controllability for the original system in the natural (L²)² norms and also in much stronger norms. This work has been done in collaboration with Sylvain Ervedoza.

Since the development of the laser some 40 years ago, a long standing dream has been to utilize this special source of radiation to manipulate dynamical events at the atomic and molecular scales. Hints that this goal may become a reality began to emerge in the 1990's, due to a confluence of concepts and technologies involving (a) control theory, (b) ultrafast laser sources, (c) laser pulse shaping techniques, and (d) fast pattern recognition algorithms. These concepts and tools have resulted in a high speed instrument configuration capable of adaptively changing the driving laser pulse shapes, approaching the performance of thousands of independent experiments in a matter of minutes. Each particular shaped laser pulse acts as a "Photonic Reagent" much as an ordinary reagent would at the molecular scale. Although a Photonic Reagent has a fleeting existence, it can leave a permanent impact. Current demonstrations have ranged from manipulating simple systems (atoms) out to the highly complex (biomolecules), and applications to quantum information sciences are being pursued. In all cases, the fundamental concept is one of adaptively manipulating quantum systems. The principles involved will be discussed, along with the presentation of the state of the field.

The talk will report on a parametrization of the real symplectic group in four dimensions. One feature of this parametrization is that it yields the polar decomposition of a symplectic matrix via the solution of simple quadratic equations. Applications to the study of squeezing transformations will be presented. Extensions to higher dimensions will be discussed.

In this joint work with Mazyar Mirrahimi, we consider an ensemble of quantum systems described by a density matrix, solution of a Lindblad-Kossakowski differential equation. We focus on the special case where the decoherence is only due to a highly unstable excited state and where the spontaneously emitted photons are measured by a photo-detector. We propose a systematic method to eliminate the fast and asymptotically stable dynamics associated to the excited state in order to obtain another differential equation for the slow part. We show that this slow differential equation is still of Lindblad-Kossakowski type, that the decoherence terms and the measured output depend explicitly on the amplitudes of quasi-resonant applied field, i.e., the control. Beside a rigorous proof of the slow/fast (adiabatic) reduction based on singular perturbation theory, we also provide a physical interpretation of the result in the context of coherence population trapping via dark states and decoherence-free subspaces. Numerical simulations illustrate the accuracy of the proposed approximation for a 5-level systems.

Hamiltonian engineering has been shown to be a powerful technique, which can be applied to many different problems that involve steering a quantum system to achieve a desirable outcome, and a particularly promising approach to Hamiltonian engineering is the optimal control approach, i.e., formulating the problem as an optimization problem. However, the problem formulation is important, and although optimization is a well-established field, the solution of the resulting optimization problems is usually not trivial, in part because the search space is usually infinite dimensional. To overcome this obstacle the controls must be parametrized, and the parametrization is critical. The most common approach is to approximate the controls using piecewise constant functions. While adequate for some problems, such a parametrization inevitably leads to high bandwidth solutions due to the discontinuities of the fields. We demonstrate that using more natural parameterizations we can significantly reduce the bandwidth of the fields, although at the expense of having to solve more complex optimization problems. Another crucial variable is the problem formulation itself. Often, optimal control problems are formulated using Hamiltonians that incorporate many approximations, e.g., RWA, off-resonant excitations and fixed couplings negligible, etc, which inevitably limit what can be achieved by optimal control. We show that we can in principle speed up the implementation of quantum gates several orders of magnitude compared to conventional frequency-selective geometric control pulses for certain systems by avoiding such approximations and taking advantage of the full range of off-resonant excitations and couplings available in the optimal control framework. Another problem with Hamiltonian engineering is that the most effective approaches are model-based, i.e., we require a model of the system, especially its response to external fields, or the functional dependence on the controls. In some cases this isn't a problem and optimal controls can be designed to be robust with regard to model uncertainties. For other problems, however, such as information transfer through spin networks using simple local actuators, it can be shown that the optimal switching sequences are highly model-dependent, while the exact network topology and precise couplings for such systems are usually not known. Such problems call for closed loop optimization. We show that we can effectively solve problems such as finding optimal switching time sequences for such networks by adapting gradient-based optimization algorithms even for problems where the standard evolutionary algorithms fail completely to find acceptable solutions. Finally, there are certain types of problems that Hamiltonian engineering, although an extremely powerful tool for quantum engineering, cannot solve. One such problem is stabilization in the presence of environmental interactions. This problem can in principle be addressed using reservoir engineering. We consider a variant of Markovian reservoir engineering using direct feedback from an indirect measurement such as homodyne detection. We show that if the control and feedback Hamiltonians in this setting are unrestricted and we have some degree of control over the type of measurement we can perform, then any state can be in principle be stabilized.

In this joint work with Elena Shchepakina we consider a canard trajectory (in the case of scalar slow variable) and a black swan (in the case of vector slow variable) as the result of gluing attractive and repulsive slow integral manifolds, due to the availability of an additional parameter (function in the case of vector slow variable) in the differential system. As a result we obtain the continuous attractive/repulsive slow invariant surface. It is possible to consider the gluing parameter (function) as a special kind of partial feedback control, which guarantees the safety of chemical regimes, even with perturbations, during a chemical process.

In this joint work with Elena Shchepakina we use a geometric singular perturbations method for reducing the model order in chemical kinetics problems. The method relies on the theory of integral manifolds, which essentially replaces the original system by another system on an integral manifold with dimension equal to that of the slow subsystem. Explicit, implicit and parametric representations of a slow invariant manifolds are used.

The powerful techniques of Optimal Control Theory (OCT), used in recent years to design laser pulse sequences to control chemical bond breaking, are applied to the problem of laser cooling in an open system. The result is a striking new mechanism in which spontaneous emission builds coherences between all the populated levels creating a pure state, only at the end of the process transferring the amplitude to the lowest energy state. This novel mechanism accelerates the cooling process by exploiting the cooling induced by spontaneous emission to all the ground electronic state levels, not just the lowest level. The mechanism suggests the calibration of cooling in terms of increasing purity of the system as measured by the quantity Tr(rho2). An analytical theory of the cooling mechanism is developed in terms of a two-stage interplay between the control fields and the spontaneous emission. One of the main results of the analytical theory is a differential equation for the optimal cooling rate. The key components of the theory – the definition of cooling as purity increase; the invariance of purity to control fields; and the maximum rate of approach to absolute zero – correspond to the zeroth, second and third law of thermodynamics, providing a thermodynamic framework for laser cooling. The formulation of cooling in terms of the coherence measure Tr(rho2) has an additional, interesting implication: that our results carry over immediately to the problem of control of quantum decoherence, suggesting both a new mechanism and fundamental limitations on the control of that process.

Decoherence, which is caused due to the interaction of a quantum system with its environment plagues all quantum systems and leads to the loss of quantum properties that are vital for quantum computation and quantum information processing. In this work we propose a novel strategy using techniques from systems theory to completely eliminate decoherence and also provide conditions under which it can be done so. A novel construction employing an auxiliary system, the bait, which is instrumental to decoupling the system from the environment, is presented. This corresponds to the Internal Model Principle for Quantum Mechanical Systems. Almost all the earlier work on decoherence control employ density matrix and stochastic master equations to analyze the problem. Our approach to decoherence control involves the bilinear input affine model of quantum control system which lends itself to various techniques from classical control theory, but with non-trivial modifications to the quantum regime. The elegance of this approach yields interesting results on open loop decouplability and Decoherence Free Subspaces (DFS). Additionally, the feedback control of decoherence may be related to disturbance decoupling for classical input affine systems, which entails careful application of the methods by avoiding all the quantum mechanical pitfalls. The two concepts are contrasted and an improved theory of disturbance decoupling for general input affine systems is developed. In the process of calculating a suitable feedback the system has to be restructured due to its tensorial nature of interaction with the environment, which is unique to quantum systems. Finally the results are also shown to be superior to the ones obtained via master equations. In order to apply feedback a reliable information extraction scheme using continuous indirect measurements with the help of a quantum probe is outlined. Finally, a methodology to synthesize feedback parameters itself is given, that technology permitting, could be implemented for practical 2-qubit systems to perform decoherence free Quantum Computing.

Major League Baseball is a multi-billion dollar per year industry that relies heavily on the quality of its schedule. Teams, fans, TV networks, and even political parties (in a way revealed in the talk) rely on the schedule for profits and enjoyment. Only recently have the computational tools of operations research been powerful enough to address the issue of finding "optimal" schedules. Trick will discuss his experiences in scheduling college basketball, major league baseball, and other sports, and show how operations research is revolutionizing the way sports scheduling is done.

The talk will begin with an introduction to the quantum master equation (Liouville-von Neumann equation), followed by a discussion of how we have used this equation it in a semiclassical algorithm for calculating of non-Born-Oppenheimer molecular dynamics. I will also discuss the physical origin of decoherence in electronically nonadiabatic molecular dynamics and our method for estimating the decoherence time. The resulting treatment will be validated against accurate quantum dynamics for small molecular systems.

In a topological space, a family of continuous mappings is called universal if its action, in at least one element of the space, is dense. If the mappings are unitary or trace-preserving completely positive, the notion of universality is closely related to the notion of controllability in either closed or open quantum systems. Quantum controllability in infinite dimensions is discussed in this setting and minimal generators are found for full control universal families. Some of the requirements of the operators needed for control in infinite dimensions follow from the properties of the infinite unitary group. Hence, a brief discussed of this group and their appropriate mathematical spaces is also included.

Joint work with Jinshuang Jin.

We present a hierarchical equations-of-motion (HEOM) formalism of quantum dissipation theory [J. Chem. Phys. 128, 234703 (2008)], which is formally exact, practically tractable, and numerically convergent. It characterizes the transient current transport dynamics of arbitrary dissipative many-electron systems, in contact with electrodes under arbitrary temperatures and external fields. The HEOM approach provides a useful theoretical tool to study various transient and stationary properties of many-body systems far away from equilibrium. With an efficient hybrid scheme accounting for the bath correlation functions, we demonstrate accurate transient response current driven by time-dependent applied voltages in both sequential and cotunneling regimes.

We have recently developed a hierarchical equations-of-motion (HEOM) approach to nonperturbative and non-Markovian quantum dissipation. It is a unified and exact theory for arbitrary coupling Gaussian environments of distinct nature: bosonic versus fermionic, and canonical versus grand canonical ensembles. It admits also an arbitrary time-dependent external field driving. Two systems will be used to elaborate both the formulation and implementation aspects of the theory. In an electron transfer (ET) system, the bath environment serves as a canonical bosonic ensemble, responsible for the system decoherence and energy relaxation. The validation of Zusman equation will be discussed, on the basis of exact HEOM results. In a quantum transport setup, a molecule or quantum dot is placed in contact with electrodes under applied voltage. Each electrode reservoir serves as a grand canonical fermion ensemble. It is responsible not only for decoherence and energy relaxation, but also for the fermion particle (i.e., electron) transport in/out of the system. The HEOM-based quantum transport theory will be summarized, together with the calculated transient currents through model quantum dot systems and the current spectrums in response to various forms of external time-dependent applied voltage.

Support from RGC of Hong Kong Government is acknowledged.

R.X. Xu and Y. J. Yan, Phys. Rev. E, 75, 031107 (2007). J. S. Jin, X. Zheng, and Y. J. Yan, J. Chem. Phys. 128, 234703 (2008). X. Zheng, J. S. Jin, and Y. J. Yan, J. Chem. Phys. 129, 184112 (2008). X. Zheng, J. S. Jin, and Y. J. Yan, New J. Phys. 10, 093016 (2008).

In this lecture we shall present a survey of recent work on several topics related with numerical approximation of waves.

Control Theory is by now and old subject, ubiquitous in many areas of Science and Technology. There is a quite well-established finite-dimensional theory and many progresses have been done also in the context of PDE (Partial Differential Equations). But gluing these two pieces together is often a hard task from a mathematical point of view.

This is not a merely mathematical problem since it affects modelling and computational issues. In particular, the following two questions arise: Are finite-dimensional and infinite-dimensional models equally efficient from a control theoretical point of view? Are controls built for finite-dimensional numerical schemes efficient at the continuous level?

In this talk we shall briefly analyze these issues for the wave equation as a model example of propagation without damping. We shall show that high frequency spurious oscillations may produce the divergence of the most natural numerical schemes. This confirms the fact that finite and infinite-dimensional modelling may give completely different results from the point of view of control. We shall then discuss some remedies like filtering of high frequencies, multi-grid techniques and numerical viscosity.

Similar questions arise when building numerical approximation schemes for nonlinear Schrödinger equations or in other contexts as when designing, for instance, absorving boundary conditions or developping the method of Perfectly Matched Layers (PML) for the wave equation.

The manipulation and control of quantum systems is fundamental to a host of emerging applications from the design of qubits and novel nanoscale devices, to the control of photochemical reactions as well as atomic and molecular dynamics. Although there are established techniques to simulate the evolution of a quantum system, the problem of finding the control potential which results in a desired evolution is considerably more challenging. Recent contributions to the development of new quantum control methodologies and optimal control formulation are discussed. In particular, the investigation of theoretical issues such as the appropriate choice of function spaces for the control and the non-convex structure of the optimization problems as well as the interplay between discretization and optimization are considered. Accurate and computationally efficient algorithms for computing the optimal controls which take advantage of the underlying physics are introduced with a focus on Krylov-Newton methods for solving controls for fast state transitions in a system.